# DFT Linearity

Explore the linearity property of the Fourier transform.

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Linearity implies that the DFT of the sum of input signals is the sum of the DFTs of the individual input signals. Let’s find out how.

## Derivation

Assume that two input signals $x_1[n]$ and $x_2[n]$ have DFTs given by $X_1[k]$ and $X_2[k]$, respectively. If that’s the case, the DFT of their sum is equal to the sum of their individual DFTs.

\begin{align*} \text{DFT }x_1[n]\quad \rightarrow\quad X_1[k]\\ \text{DFT }x_2[n]\quad \rightarrow\quad X_2[k] \end{align*}

This implies that:

$\text{DFT }\Big\{y[n]=x_1[n]+x_2[n]\Big\}\quad \rightarrow\quad Y[k]=X_1[k]+X_2[k]$

This is because:

\begin{align*} Y[k]&=\sum_{n=0}^{N-1}\Big\{x_1[n]+x_2[n]\Big\}e^{-j2\pi\frac{k}{N}n}\\ &=\sum_{n=0}^{N-1}x_1[n]e^{-j2\pi\frac{k}{N}n}+\sum_{n=0}^{N-1}x_2[n]e^{-j2\pi\frac{k}{N}n}\\ &=X_1[k]+X_2[k] \end{align*}

This helps determine the DFT of a signal without explicit calculations.

## Example

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